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It refers to the girl. "That one" is a somewhat dated construction, and expresses mild derision or mild admiration. All the examples I could think of sounded just like yours: "She/he is [adjective or noun-phrase], that one." Insert something like "a handful", "a beaut", "one in a million", "a piece of work".

The terms "that", "then", and "that one", heard after a statement and referring to its subject, are still commonly heard in British English. There are similar constructs in French speech ("c'est chouette, ca" literally translates as "that's cool, that") which, given Britain's history, makes me think there's a relation, but American English has mostly dropped this particular redundancy.
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KeithSJul 20 '11 at 16:32

@KeithS -- do actual Brits pop "what" and "that" into sentences for emphasis or is that just bad American parodies? I have the same question about calling people "old man" and "guv'nor".
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MalvolioJul 20 '11 at 20:09

Not as often as our bad American parodies would indicate, but it is present in B.E. Because it is conspicuously absent in A.E. we notice its presence in B.E., and it becomes the basis for stereotype. However, the example in the OP is from a contemporary British author of pop fiction (J.K. Rowling), and although she does lampoon various British and European cultures and sub-cultures, Hermione, being a down-to-earth braniac, would not be a character to use to lampoon British speech forms.
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KeithSJul 20 '11 at 20:11